Rectangular section - Compound bending (SLS)

Rectangular section - Compound bending (SLS)SymbolsGeometryForcesContraintesSLS check principleCalculation assumptionsModes of behavior of the sectionSLS reinforced concrete section checkCalculation of reinforcement sectionPartially and fully compressedSimple tractionCrack opening ( $w_k$ )Bibliography

Symbols

Geometry

Symbol	Unit	Description
b	m	Width of the section
h	m	Total height of the section
$A_{s1}$	m²	Lower steel section
$A_{s2}$	m²	Upper steel section
$c_1$	m	Distance between the top of the section and the center of gravity of the lower steels
$c_2$	m	Distance between the bottom of the section and the center of gravity of the upper steels
$d_2$	m	$A_{s2}$ in relation to the lower fiber of the section
$v’$	m	Position of G' with respect to the lower fiber of the section
G	-	Center of gravity of the rectangular section
G’	-	Center of gravity of the homogenized section

Forces

Symbol	Unit	Description
$N_s$	MN	Normal force applied to G
$M_s$	MNm	Bending moment applied to G
$M$	MNm	Equivalent bending moment applied to the lower fibre of the section
$N$	MN	Equivalent normal force applied to the lower fibre of the section
$M'$	MNm	Equivalent bending moment applied to G''
$N'$	MN	Equivalent normal force applied to G''
$F_{s1}$	MN	Equivalent normal force applied to G''
$F_{s2}$	MN	Force taken up by the compressed reinforcement

Contraintes

Symbol	Unit	Description
$\sigma_{s1}$	MN/m²	Stress in tensioned steels
$\sigma_{s2}$	MN/m²	Stress in compressed steels
$\sigma_c$	MN/m²	Compressive stress of concrete
$f_{ck}$	MN/m²	Compressive stress of concrete (characteristic value)
$f_{yd}$	MN/m²	Allowable compression/tension stress of steel (design value)
$f_{yk}$	MN/m²	Stress of the steel (characteristic value)

SLS check principle

The SLS check is performed using a stress based analysis.

This stress based analysis requires a calculation on an homogenized cross section considering the different deformation moduli of steel and concrete.

Calculation assumptions

The SLS check of reinforced concrete sections takes into account the following hypotheses:

Hypothesis 1 : The tensile strength of concrete is neglected.
Hypothesis 2 : $\overline{\sigma_c}$ at SLS.

Hypothesis 3 : $\overline{\sigma_s}$ at SLS.

The constitutive laws of concrete and steel are provided in the Materials chapter of this manuals.

Modes of behavior of the section

$M_s$ $N_s$ ), the reinforced concrete section can be in:

Simple compression: normal compression force and no moment.

Stress diagram of a simple compressed section

Partially compressed: normal compression force with moment

Stress diagram of a partially compressed section

Fully compressed: normal compression force with moment.

Stress diagram of a fully compressed section

Fully tensioned: normal tensile force with moment

Stress diagram of a fully tensioned section

SLS reinforced concrete section check

The verification of the reinforced concrete section with SLS is performed by homogenizing the section in order to take into account the presence of two materials with different stiffnesses. The equilibrium of the section is based on the equilibrium of forces and moments, which makes it possible to obtain the stress diagram of the section.

In the case of a fully tensioned section, concrete does not provide any tensile strength. Only the reinforcement forces balance the external forces.

Calculation of reinforcement section

Partially and fully compressed

The principle of dimensioning in SLS consists in looking for the minimum cross section verifying the equilibrium of the cross section while guaranteeing that the limit stresses of each material are not exceeded.

Simple traction

In simple traction, the tensile strength of concrete is neglected. Only the reinforcement forces counterbalance the forces applied to the section. The most economical solution is to guarantee that the center of gravity of the reinforcement is at the point of application of normal force.

Let's note:

$e_{s1}$ distance between the point of application of the normal force and the tensioned reinforcement
$e_{s2}$ distance between the point of application of the normal force and the compressed reinforcement

The steel sections are obtained from the moments equilibrium:

A_{s2}=\frac{Ne_{s1}}{\left(e_{s1}+e_{s2}\right)\sigma_{s2}}

A_{s1}=\frac{Ne_{s2}}{\left(e_{s1}+e_{s2}\right)\sigma_{s1}}

The stresses in steels are considered equal to the allowable stress in the SLS.

$w_k$ )

w_k=s_{r,max}\left(\varepsilon_{sm}-\varepsilon_{cm}\right)

Symbol	Unit	Description
$w_k$	mm	Opening cracks
$s_{r,max}$	mm	Maximum crack spacing
$\varepsilon_{sm}$	-	Average reinforcement elongation, under the combination of actions considered, taking into account the contribution of tensioned concrete
$\varepsilon_{cm}$	-	Average concrete elongation between cracks

s_{r,max}=k_3c+0.425k_1k_2\frac{\phi_{eq}}{\rho_{p,eff}}

Symbol	Unit	Description
c	mm	Coating of longitudinal reinforcement
$k_1$	-	Coefficient function of the adhesion properties of the bars (0.8 for HA bars)
$k_2$	-	$k_2 = 0.5$ $k_2 = 1$ $k_2=\frac{\varepsilon_1+\varepsilon_2}{2\varepsilon_1}$ $\varepsilon_1$ $\varepsilon_2$ the lowest of the relative elongations of the relative elongations of the fibers and the cross-section considered, evaluated on the basis of a cracked cross-section.
$k_3$	-	Coefficient of coating adjustment, equal to 3.4
$\phi_{eq}$	mm	Equivalent diameter of the bars
$\rho_{p,eff}$	-	$A_s/A_{c,eff}$ )
$A_{c,eff}$	m²	$h_{c,ef}=min\left\{2.5\left(h-d\right)\;;\;\left(h-x\right)/3\;;\;h/2\right\}$

\varepsilon_{sm}-\varepsilon_{cm}=max\left\{0.6\frac{\sigma_s}{E_s};\frac{\sigma_s+k_t{\displaystyle\frac{f_{ct,eff}}{\rho_{p,eff}}}\left(1+\alpha_e\rho_{p,eff}\right)}{E_s}\right\}

Symbol	Unit	Description
$\sigma_s$	MN/m²	SLS stress on tensioned reinforcement, calculated by assuming the cracked section
$E_s$	MN/m²	Elastic modulus of steel
$f_{ct,eff}$	MN/m²	$f_{ct,eff}=f_{ctm}$ )
$k_t$	-	$k_t=0.6$ $k_t=0.4$ for long term loading

Bibliography

Eurocode 2
Calculation in simple or compound bending at the ultimate limit state of the rectangular reinforced concrete sections (A. CAPRA, M. HAUTCOEUR)
Calculation of reinforced concrete structures (Perchat)